Traditionally, unitarity of the scattering matrix is implemented at
the integrated level via dispersion relations (see
e.g. Ref. [1]}). However, for our purposes, it is
much more convenient to use an integrand-level version of unitarity [2]}.
This is based on the concept of a generalized-unitarity cut that reduces an integrand
to a sum of products of tree-level amplitudes. For example, for the
\(s\) -channel cut displayed in Fig. REF (a),
\(\mathcal {M}_4^\text{1-loop} \Big \vert _{s\rm \hbox{-}cut} \!\!= \! \int \!\!\frac{d^{4-2\epsilon } L}{(2\pi )^{4-2\epsilon }} \frac{i}{L_1^2-m^2}\frac{i}{L_2^2-m^2} \sum _{\rm states} \! \mathcal {M}_4^{\rm tree}(1, 2, L_1,L_2) \mathcal {M}_4^{\rm tree}(-L_1, -L_2, 3, 4)\Big \vert _{s\rm \hbox{-}cut} ,\) 
